A Hermitian operator is a mathematical operator in linear algebra that has the property of being self-adjoint. This means that the operator is equal to its own conjugate transpose, and therefore has real eigenvalues.
Some common examples of Hermitian operators include the position and momentum operators in quantum mechanics, as well as the Hamiltonian operator which represents the total energy of a quantum system.
Some key properties of Hermitian operators include having real eigenvalues, being self-adjoint, and having orthogonal eigenvectors. They also have the property that the expectation value of a Hermitian operator is always a real number.
An operator is Hermitian if it is equal to its own conjugate transpose, meaning that the operator must satisfy the condition A = A†. This can be checked by computing the conjugate transpose of the operator and comparing it to the original operator.
Hermitian operators are fundamental in quantum mechanics as they represent physical observables of quantum systems. They are used to calculate the probabilities of various outcomes in experiments, as well as to determine the time evolution of quantum states.